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Harmonic Oscillators:

 

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Vibe:Harmonic Oscillators:Damped Mass Spring Oscillator

This oscillator is identidal to the simple Mass Spring save one addition: it is damped using a damping factor zeta. Here is the second-order differential equation for the damped mass spring:

The damping factor zeta is in fact the ratio of actual damping to that of critical damping. This equation represents the motion of a mass constrained by a linear elastic spring in parallel wiht a dashpot full fo oil, which is assumed to provide a force opposing the instantaneous velocity. When zeta is >1 we get critical damping and the system becomes a point-attractor. That is, the system doens't oscillate at all. When zeta is <0 the system becomes a repeller and the oscillator diverges.

Vibe Hints and Experiments
  • First, click on the Launch Simulator! button in the upper left. If your browser supports Java, you will get the Vibe simulator in a separate window. If you don't see a Launch Simulator button, you are either browsing Vibe with Java turned off or, alternately, Java is not working on your browser.
  • Now run the oscillator with default settings for a few timesteps.
  • Click the Layer Button to run a second simulation
  • This time try increasing damping to a value greater than 1 and then click Start to run the new simulation for a few timesteps. This system should damp so well that it doesn't oscillate at all. Rather the system moves directly to a single point at (0,0); it is a point attractor
  • Now click the Layer Button again to run a third simulation
  • This time try setting damping to a small negative number, say -2 and then click Start to run the new simulation. This time the system should repel due to the negative damping.
  • Don't be confused by the constant resetting of the display windows. This is normal. Why? As the oscillator repels further and further away from (0,0). Vibe automatically zooms out.

Vibe Oscillator Simulator © 2000-2003 Douglas S. Eck.
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Thu Jul 24 09:12:34 EDT 2003